Averages question

Question

There are five sticks. The average length of any four of them is 600. What is the average of all five?

Is it possible to find the average of all with just this information given?

Answer 1

Let the lengths of the five sticks be $a_i,1\le i\le 5$

So, $\displaystyle a_1+a_2+a_3+a_4=4\cdot600=2400 \ \ \ \ (1)$

$\displaystyle a_1+a_2+a_3+a_5=4\cdot600=2400 \ \ \ \ (2)$

$\cdots$

Subtracting we get $a_4=a_5$

Similarly, we can show that the length of all the sticks are same.

I should not say anything more:)

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Alternatively,

Clearly, there are $\binom 54=5$ combinations

Observation each of $a_i$ occurs $5-1$ times in the combinations

Adding we get $(5-1)(a_1+a_2+a_3+a_4+a_5)=5\cdot2400$